1. Putting Laziness to Work

2. Why use laziness Laziness has lots of interesting uses – Build cyclic structures. Finite representations of infinite data. – Do less work, compute only those values demanded by the final result. – Build infinite intermediate data structures and actually materialize only those parts of the structure of interest. Search based solutions using enumerate then test. – Memoize or remember past results so that they don’t need to be recomputed

3. Cyclic structures cycles:: [Int] cycles = 1 : 2 : 3 : cycles 123

4. Cyclic Trees data Tree a = Tip a | Fork (Tree a) (Tree a) t2 = Fork (Fork (Tip 3) (Tip 4)) (Fork (Tip 9) t2) Fork Tip 3Tip 4Tip 9

5. Mutually Cyclic (t3,t4) = ( Fork (Fork (Tip 11) t3) t4, Fork (Tip 21) (Fork (Tip 33) t3) ) Fork Tip 11 Tip 21 Fork Tip 33

6. Prime numbers and infinite lists primes :: [Integer] primes = sieve [2..] where sieve (p:xs) = p : sieve [x | x<-xs, x `mod` p /= 0]

7. Dynamic Programming Consider the function fib :: Integer -> Integer fib 0 = 1 fib 1 = 1 fib n = fib (n-1) + fib (n-2) LazyDemos> :set +s LazyDemos> fib 30 1346269 (48072847 reductions, 78644372 cells, 1 garbage collection) takes about 9 seconds on my machine!

8. Why does it take so long? Consider (fib 6) fib 6 fib 5 fib 4 fib 2 fib 1fib 0 fib 3 fib 2 fib 1fib 0 fib 1 fib 3 fib 2 fib 1fib 0 fib 1 fib 4 fib 2 fib 1fib 0 fib 3 fib 2 fib 1fib 0 fib 1

9. What if we could remember past results? Strategy – Create a data structure – Store the result for every (fib n) only if (fib n) is demanded. – If it is ever demanded again return the result in the data structure rather than re-compute it Laziness is crucial Constant time access is also crucial – Use of functional arrays

10. Lazy Arrays import Data.Array table = array (1,5) [(1,'a'),(2,'b'),(3,'c'),(5,'e'),(4,'d')] The array is created once Any size array can be created Slots cannot be over written Slots are initialized by the list Constant access time to value stored in every slot

11. Taming the duplication fib2 :: Integer -> Integer fib2 z = f z where table = array (0,z) [ (i, f i) | i <- range (0,z) ] f 0 = 1 f 1 = 1 f n = (table ! (n-1)) + (table ! (n-2)) LazyDemos> fib2 30 1346269 (4055 reductions, 5602 cells) Result is instantaneous on my machine

12. Can we abstract over this pattern? Can we write a memo function that memoizes another function. Allocates an array Initializes the array with calls to the function But, We need a way to intercept recursive calls

13. A fixpoint operator does the trick fix f = f (fix f) g fib 0 = 1 g fib 1 = 1 g fib n = fib (n-1) + fib (n-2) fib1 = fix g

14. Generalizing memo :: Ix a => (a,a) -> ((a -> b) -> a -> b) -> a -> b memo bounds g = f where arrayF = array bounds [ (n, g f n) | n <- range bounds ] f x = arrayF ! x fib3 n = memo (0,n) g n fact = memo (0,100) g where g fact n = if n==0 then 1 else n * fact (n-1)

15. Representing Graphs import ST import qualified Data.Array as A type Vertex = Int -- Representing graphs: type Table a = A.Array Vertex a type Graph = Table [Vertex] -- Array Int [Int] Index for each node Edges (out of) that index 1 23 45 7 8 9 10 6 1[2,3] 2[7,4] 3[5] 4[6,9,7] 5[8] 6[9] 7 8[10] 9 10[]

16. Functions on graphs type Vertex = Int type Edge = (Vertex,Vertex) vertices :: Graph -> [Vertex] indices :: Graph -> [Int] edges :: Graph -> [Edge]

17. Building Graphs buildG :: Bounds -> [Edge] -> Graph graph = buildG (1,10) [ (1, 2), (1, 6), (2, 3), (2, 5), (3, 1), (3, 4), (5, 4), (7, 8), (7, 10), (8, 6), (8, 9), (8, 10) ] 1 2 5 6 4 7 8 9 10 3

18. DFS and Forests data Tree a = Node a (Forest a) type Forest a = [Tree a] nodesTree (Node a f) ans = nodesForest f (a:ans) nodesForest [] ans = ans nodesForest (t : f) ans = nodesTree t (nodesForest f ans) Note how any tree can be spanned by a Forest. The Forest is not always unique. 1 2 5 6 4 7 8 9 10 3

19. DFS The DFS algorithm finds a spanning forest for a graph, from a set of roots. dfs :: Graph -> [Vertex] -> Forest Vertex dfs g vs = prune (A.bounds g) (map (generate g) vs) generate :: Graph -> Vertex -> Tree Vertex generate g v = Node v (map (generate g) (g `aat` v)) Array indexing An infinite cyclic tree

20. Sets of nodes already visited import qualified Data.Array.ST as B type Set s = B.STArray s Vertex Bool mkEmpty :: Bounds -> ST s (Set s) mkEmpty bnds = newSTArray bnds False contains :: Set s -> Vertex -> ST s Bool contains m v = readSTArray m v include :: Set s -> Vertex -> ST s () include m v = writeSTArray m v True Mutable array

21. Pruning already visited paths prune :: Bounds -> Forest Vertex -> Forest Vertex prune bnds ts = runST (do { m <- mkEmpty bnds; chop m ts }) chop :: Set s -> Forest Vertex -> ST s (Forest Vertex) chop m [] = return [] chop m (Node v ts : us) do { visited <- contains m v ; if visited then chop m us else do { include m v ; as <- chop m ts ; bs <- chop m us ; return(Node v as : bs) }

22. Topological Sort postorder :: Tree a -> [a] postorder (Node a ts) = postorderF ts ++ [a] postorderF :: Forest a -> [a] postorderF ts = concat (map postorder ts) postOrd :: Graph -> [Vertex] postOrd = postorderF. Dff dff :: Graph -> Forest Vertex dff g = dfs g (vertices g)

23. A BC DE G H I J F A BC DE G H I J F (0,””) (1,A) (5,A) (2,CA) (3,ECA) (4,HECA) (6,BA) (7,DBA) (8,GDBA) (9,DBA) A. Control Flow Graph a :: Graph Char a B. DFS Labeled Graph b :: Graph Char (Int,[Char]) dfsnum :: v->Int dfsnum v = fst(apply b v) dfspath:: v->[v] dfspath v = snd(apply b v) A BC DE G H I J F (A)(A) (A)(A) (A)(A) (C)(C) (C)(C) (A)(A) (B)(B) (B)(B) (D)(D) (D)(D) C. Semi-Dominator Labeled Graph c :: Graph Char Char semi :: v->v semi = apply c A BC D GE J H FI D. Semi-Dominator Graph d :: Graph Char a A BC D GE J H F I E. Dominator Graph e :: Graph Char a dfslabel f map g induce